"If you plan on picking up the LittleBigPlanet 2 Cross-Controller Pack from the PlayStation Store sometime this week then be sure to download the game’s latest patch that was released earlier today."
(LittleBigPlanet 2, PS Vita, PS3)

No, the patch only has to be installed on the PS3. All the content is in the Patch for the PS3 and there has been no mention whatsoever from the devs that you would need the same account. The Vita just becomes the controller, not the client. You can also play the content even if you don't have the DLC, so ANY Vita should work.

Don't forget that you can have more than one Vita used in the same game, so that clearly proves you don't need the same account.